Deltoidal hexecontahedron

Deltoidal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Sladit
Coxeter diagram	m5o3m
Elements
Faces	60 kites
Edges	60+60
Vertices	12+20+30
Vertex figure	20 triangles, 30 squares, 12 pentagons
Measures (edge length 1)
Dihedral angle	$\arccos\left(-\frac{19+8\sqrt5}{41}\right) ≈ 154.12136°$
Central density	1
Related polytopes
Army	Sladit
Regiment	Sladit
Dual	Small rhombicosidodecahedron
Conjugate	Great deltoidal hexecontahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

The deltoidal hexecontahedron, also called the strombic hexecontahedron or small lanceal ditriacontahedron, is one of the 13 Catalan solids. It has 60 kites as faces, with 12 order-5, 20 order-3, and 30 order-4 vertices. It is the dual of the uniform small rhombicosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron, an icosahedron, and an icosidodecahedron. If the dodecahedron has unit edge length, the icosahedron's edge length is $\frac{7+\sqrt5}{6} ≈ 1.53934$ and the icosidodecahedron's edge length is $\frac{4-\sqrt5}2 ≈ 0.88197$.

Each face of this polyhedron is a kite with its longer edges $\frac{7+\sqrt5}{6} ≈ 1.53934$ times the length of its shorter edges. These kites have one angle measuring $\arccos\left(-\frac{5+2\sqrt5}{20}\right) ≈ 118.26868°$, the opposite angle measuring $\arccos\left(\frac{9\sqrt5-5}{40}\right) ≈ 67.78301°$, and the other two angles measuring $\arccos\left(\frac{5-2\sqrt5}{10}\right) ≈ 86.97416°$.

Related polytopes

The deltoidal hexecontahedron is topologically equivalent to the rhombic hexecontahedron which has golden rhombi for faces instead of kites.

External links

• Klitzing, Richard. "Sladit".

• Wikipedia Contributors. "Deltoidal hexecontahedron".
• McCooey, David. "Deltoidal Hexecontahedron"

• Hi.gher.Space Wiki Contributors. "Deltoidal hexecontahedron".